Concentration Fluctuations in Turbulent Diffusion

A conservation law is derived from the equation of continuity for the mean-square concentration fluctuations (msf) in turbulent diffusion. The terms of this law are very similar to those found in the energy balance of a turbulent mixing layer or boundary layer. A closed equation for the msf in a continuous plume in a uniform field of turbulence is obtained by the following assumptions: a) that the flux of both concentration and msf may be expressed as an exchange coefficient times gradient; b) that the two exchange coefficients involved are equal and constant for a given cross section of the plume, although they vary with distance from the source; c) and that the dissipation rate of msf is proportional to the local msf and that the constant of proportionality, a “decay-time scale,” is also constant for a given section (even though it is a function of distance). Self-similar profiles of msf are found to be possible, provided that the decay-time scale varies inversely with distance from the source. Comparison with experiment shows that profiles of msf are self-similar, implying a variation of T with x−1. Experimental and calculated theoretical profiles of msf show fair agreement. An application of these results to the prediction of “peak-to-mean ratios” offers a rational explanation of such ratios observed in the atmosphere.